Secrets of Creation, Volume One: The Mystery of the Prime Numbers is the first of a trilogy originally intended to be a single book. The subject of this book is one of the oldest and most beautiful in mathematics. However, unlike the numerous popular books that have probed deeply into the beauty and mystery of the prime numbers and the Riemann Hypothesis (such as the ones by M. du Sautoy, J. Derbyshire, etc.), this one is unique in many aspects. It is the only book geared towards general audience that actually gives an account of the complexity of the distribution of the prime numbers while containing almost no equations.
So, how did the author achieve this? Through the radiant wit of the illustrations paired up with the exceptional writing. Additionally, his success is due to the simplicity of the explanations, which in an engaging manner reveal complex concepts with such clarity as to make the book accessible to children and non-mathematicians. Lastly, the exposition of the beauty of the harmonic decomposition is superb.
The book begins as the author examines the role of numbers in the modern Western world. He argues that based on their use in counting, the numbers are visualized as entities stretched out in a line. Thus, “we’ve built a civilization around this number system and yet the most fundamental fact about it is unknown to almost everyone.” Since the “number system” is vital to how we perceive reality “the aim of this book” in the author’s words “is to try to balance the view and experience of numbers by focusing on prime factorization as an accessible, indisputable yet utterly mysterious aspect of the number system.”
In order to accomplish this, the author discusses addition and multiplication-based approaches to building the number system. The first is based on how we were taught to count (by adding 1s), while the second consists of multiplying primes together. Pictorially, the first approach in the book is illustrated by placing the numbers in a sequence on a line, while in the second one the number is represented by a cluster of balloons (which represent the factors of that number). This captivating visualization of the integers shows them as more than just a counting system and exposes the mysterious relationship between the multiplication and addition, which is still not fully understood.
Next, the author sets off to crack open and reveal the internal structure (the prime factors) of the numbers instead of viewing them as dull entities distinguished only by their position on the number line. He introduces the prime numbers and looks at the number not as a quantity but rather as a unique “cluster” of primes which when multiplied together make up that number. Consequently, Watkins starts a quest to find a pattern in the irregular distribution of primes in the sequence of numbers.
Mathematicians have tried to find ways to approximate this distribution with greater accuracy. One of the latest formulas that comes very close to this represents a collection of what in the book is described as “spiral waves”. So, what are they?
As the prime numbers increase they can be found on the uncoiling of a logarithmic spiral in a way that could not be predicted, but yet follows a very precise pattern. Watkins takes the standard natural log (which is so closely related to the general distribution of the primes) and turns it into the form of an equiangular spiral. Geometrically, the equiangular spirals fuse simultaneously linear and circular characteristics in an ordered manner. Thus, the distribution of the primes can be mapped as spiral waves and it can be represented by harmonic analysis.
This book goes even further by broadening the notion of spirals into the treatment of the famous deviations in the Riemann Prime Counting Function, which are closely related to the non-trivial zeros of the Zeta Function. The deviations of the actual arrangement of prime numbers from this “approximate” pattern are related to an infinite collection of wave-like forms, explained in terms of spirals, which the author has chosen to name them spiral waves. Similar to a conventional wave (such as the “sine wave”), they have something like a “frequency”. What makes this so interesting are the mysterious irrational frequencies, such as 14.134725..., 21.022040...and 25.010856...and where they have come from.
Every argument in the book is backed up vividly by numerous illustrative examples and user-friendly visualizations. The illustrator, Matt Tweed, has provided brilliant drawings that convey and explain the mathematical ideas and concepts as they develop throughout the book. What could have been rigorous text saturated with formulas is replaced by the many entertaining illustrations which also offer an intuitive understanding of the technical complexities behind the text. For example, there is a wizard walking in a spiral while dropping a candle on each prime he passes (therefore showing how prime numbers are distributed.) A prime number is represented by number of beans which can’t be arranged in a rectangular grid. The factorization is a seen as a bowl containing different fruit. There are many characters: the wizard, the adorable little children, robots, angels, a busy scientist, piles of beans, aliens, ladybugs, and other engaged in activities that make the content of the text compelling to anyone. (My 3-year old son literally could not put this book down; he had many questions and comments about what happens in the drawings.)
Additionally, the book offers historical background as well as explanations of the role of our number system in religion, economics, philosophy and psychology. The author suggests web sites with further exploration on the subject. Moreover, he discusses the reactions and the feelings (surprise, astonishment, bafflement) of known mathematicians about mathematical truths, the prime numbers, and their distribution.
This is a very unusual and inspiring book with the potential to spark interest even among experts on prime number theory. The exciting and original presentation is instructive and stimulates further study. Diving into the mystery of numbers in this book leaves one thirsting for the subsequent two volumes.
A native of Macedonia, Ana Momidic-Reyna has an M.S. in Mathematics and has also worked for the high energy physicists at Fermilab. While waiting for the opportunity to work on her Ph.D. in mathematics, she keeps up with the field by reading as many mathematics books as she can.